Dividing a Segment into N parts:
Besteman Construction II
A Mathematical Droodle: What Is This About?

Explanation

The applet illustrates an algorithm for dividing a segment in N equal parts found by a college student, Nathan Besteman, and reported in the recent issue of Mathematics Teacher.

The construction employs a well known property of angle bisectors in a triangle. In ACD, let DP be the angle bisector of angle D. Then

It follows from (1) that if CD = N·AD, then AP = PC/N, or PC = N·AP, so that by adding AP we get AP = AC/(N+1). This suggests an iterative algorithm: for N = 2, there is a known construction of the midpoint B of AC by joining the points of intersection of two circles of equal radii centered at A and C. To trisect AC, reduce the radius of the circle at A to AB, which gives a new triangle ACD. The angle bisector of D will now divide AC in the ratio 1:2, such that AP = AC/3. Reduce the radius of the circle at A to AP and consider a new triangle ACD. The angles bisector will now divided AC in the ratio 1:3, and so on.

References

1. N. Besteman and J. Ferdinands, Another Way to Divide a Line Segment into n Equal Parts, Mathematics Teacher, Vol. 98, No. 6 (Feb. 2005), pp. 428-433

• How to divide a segment into n equal parts
• Al-Nayrizi's Construction
• Besteman's Construction
• Besteman Construction II
• Dividing a Segment by Paper Folding
• Euclid's Segment Division
• The GLaD Construction
• The SaRD Construction
• Similar Right Triangles